Quadratic Character Profile Redundancy

Abstract

Every mod-sixty quadratic character is generated by the Gaussian, Eisenstein, and golden splitting characters and is redundant on their fibers.

Theorem 1.1 (Additional quadratic characters do not refine the three-ring profile).

$$\begin{array}{c}(\forall ch{i}_{new}:\mathrm{QuadraticObserver}\left((\mathbb{Z}/60\mathbb{Z}{)}^{\times }\right),\\ \exists c:\mathrm{Fin}\left(3\right)\to \mathbb{Z}/2\mathbb{Z},\\ \forall u:(\mathbb{Z}/60\mathbb{Z}{)}^{\times },ch{i}_{new}(u)=ch{i}_G(u{)}^{\mathrm{val}\left(c(0)\right)}\cdot ch{i}_E(u{)}^{\mathrm{val}\left(c(1)\right)}\cdot ch{i}_5(u{)}^{\mathrm{val}\left(c(2)\right)})\wedge \\ (\forall ch{i}_{new}:\mathrm{QuadraticObserver}\left((\mathbb{Z}/60\mathbb{Z}{)}^{\times }\right),u,v:(\mathbb{Z}/60\mathbb{Z}{)}^{\times },\\ triRingImage(u)=triRingImage(v)\Rightarrow ch{i}_{new}(u)=ch{i}_{new}(v))\wedge \\ (\forall t:ThreeRingProfile,\Vert \{u:(\mathbb{Z}/60\mathbb{Z}{)}^{\times }\mid triRingImage(u)=t\}\Vert =2).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/Splitting/QuadraticCharacterProfileRedundancy.quadratic_characters_are_three_ring_products_and_fiber_redundant (✓ std3). ∎

Source. Repository-derived.

Commentary.

The three named characters are constructed from the existing Gaussian, Eisenstein, and golden split/inert readings. Split maps to the multiplicative identity and inert to the nonidentity binary value.

The unit classes nineteen, twenty-nine, and thirteen give dual generators for these readings. Every unit modulo sixty decomposes over those generators, so the values of an arbitrary quadratic observer determine the displayed three coefficients.

Equal three-ring profiles give equal values for each generating character and therefore for every quadratic observer. The frozen profile theorem supplies the exact cardinality two of every fiber.

References

• Truth anchor: D5/S3/PrimeForms/Splitting/QuadraticCharacterProfileRedundancy.quadratic_characters_are_three_ring_products_and_fiber_redundant
• Dependency: D5/S3/Factorization/Galois/QuadraticObservationBound
• Dependency: D5/S3/PrimeForms/Splitting/ThreeRingProfileFibers